Re: Power Law, 幂律定律

幂律分布研究简史

胡海波* 王林
（西安理工大学电子系 西安 710048）

摘 要 自然界与社会生活中存在各种各样性质迥异的幂律分布现象，因而对它们的研究具有广泛而深远的意义。近年来，借助于有效的物理和数学工具，及强大的计算机运 算能力，科学家们对幂律分布的本质有了进一步深层次的理解。本文从统计物理学的角度，简要介绍了幂律分布的研究史以及最新的进展，并对它的形成机制及动力 学影响作了一些言简意赅的阐述。

关键词 幂律分布，优先连接，自组织临界，HOT理论

A brief research history of power law distributions

HU Hai-Bo WANG Lin
(Department of Electronic Engineering, Xi'an University of Technology, Xi'an 710048, China)

Abstract: Various power law distribution phenomena with different characters are ubiquitous in nature and society, thus their research carries broad and far-reaching significance. In recent years, by effective physical and mathematical tools and powerful computational faculties, scientists have had a farther and substantial understanding of the essence of power law distributions. This paper introduces briefly the research history and current development of power law distributions from the perspective of statistical physics, and presents some concise and comprehensive expatiation on the mechanisms for generating them and their influence on certain dynamic characters.

Key words: power law distributions, preferential attachment, self-organized criticality (SOC), highly optimized tolerance (HOT)

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* E-mail: sdhuzi@163.com
§ E-mail: wanglin@xaut.edu.cn
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1 引言

　 　自然界与社会生活中，许多科学家感兴趣的事件往往都有一个典型的规模，个体的尺度在这一特征尺度附近变化很小。比如说人的身高，中国成年男子的身高绝大 多数都在平均值1.70米左右，当然地域不同，这一数值会有一定的变化，但无论怎样，我们从未在大街上见过身高低于10厘米的"小矮人"，或高于10米的 "巨人"。如果我们以身高为横坐标，以取得此身高的人数或概率为纵坐标，可绘出一条钟形分布曲线（如图1左图所示），这种曲线两边衰减地极快；类似这样以 一个平均值就能表征出整个群体特性的分布，我们称之为泊松分布。另外一个我们要注意的，是最高的人与最矮的人的身高之比，根据吉尼斯世界纪录[1, 2]，世界上最高的人与最矮的人（均已去世）的身高分别是2.72米和0.57米，二者之比为4.8，这个数值并不是很大，我们将在下文中证实。

　 　对于另一些分布，像国家GDP或个人收入的分布，情况就大不一样了，个体的尺度可以在很宽的范围内变化，这种波动往往可以跨越多个数量级。比如根据世界 银行的统计[3]，最富有的国家――自然是美国――其2003年GDP高达10,881,609,000,000美元（一个天文数字），而数据显示同年 GDP最低的国家――西非岛国圣多美和普林西比――只有54,000,000美元，二者之比高达201511.3。个人收入分布亦是如此，想想世界首富比 尔・盖茨那高达465亿美元的个人资产就清楚了。国家或城市人口的分布也会出现类似的情形，据世界银行的统计[4]，2003年人口最多的国家――中国 ――总人口数多达1,288,400,000，而数据显示同年人口最少的国家――西太平洋上的帕劳群岛――人口数仅为20,000（不及中国一个普通县城 的人口数），二者之比有64420之多。以收入或人口数为横坐标，以不低于该收入值或人口数的个体数或概率为纵坐标，可绘出一条向右偏斜得很厉害，拖着长 长"尾巴"的累积分布曲线（如图1右图所示），它与钟形的泊松分布曲线有显著的不同。这种"长尾"分布表明，绝大多数个体的尺度很小，而只有少数个体的尺 度相当大，像国家人口，全世界有300多个国家和地区，只有11个国家的人口数超过一亿。

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图1 泊松分布(左)与"长尾"分布(右)

2 幂律分布研究：上个世纪及以前

　　对"长尾"分布研究做出重要贡献的是Zipf和Pareto[5]，虽然他们并不是这种分布的最早发现者。

　 　1932年，哈佛大学的语言学专家Zipf在研究英文单词出现的频率时，发现如果把单词出现的频率按由大到小的顺序排列，则每个单词出现的频率与它的名 次的常数次幂存在简单的反比关系：P(r)～r^(-α)，这种分布就称为Zipf定律，它表明在英语单词中，只有极少数的词被经常使用，而绝大多数词很 少被使用。实际上，包括汉语在内的许多国家的语言都有这种特点。物理世界在相当程度上是具有惰性的，动态过程总能找到能量消耗最少的途径，人类的语言经过 千万年的演化，最终也具有了这种特性，词频的差异有助于使用较少的词汇表达尽可能多的语义，符合"最小努力原则"。分形几何学的创始人 Mandelbrot[6]对Zipf定律进行了修订,增加了几个参数，使其更符合实际的情形。

　　19世纪的意大利经济学家 Pareto研究了个人收入的统计分布，发现少数人的收入要远多于大多数人的收入，提出了著名的80/20法则，即20%的人口占据了80%的社会财富。 个人收入X不小于某个特定值x的概率与x的常数次幂亦存在简单的反比关系：P[X≥k]～x^(-k)，上式即为Pareto定律。

　　 Zipf定律与Pareto定律都是简单的幂函数，我们称之为幂律分布；还有其它形式的幂律分布，像名次――规模分布、规模――概率分布，这四种形式在数 学上是等价的[5, 7]，幂律分布的示意图如图1右图所示，其通式可写成y=c*x^(-r)，其中x，y是正的随机变量，c，r均为大于零的常数。这种分布的共性是绝大多 数事件的规模很小，而只有少数事件的规模相当大。对上式两边取对数，可知lny与lnx满足线性关系，也即在双对数坐标下，幂律分布表现为一条斜率为幂指 数的负数的直线，这一线性关系是判断给定的实例中随机变量是否满足幂律的依据。判断两个随机变量是否满足线性关系,可以求解两者之间的相关系数；利用一元 线性回归模型和最小二乘法可得lny对lnx的经验回归直线方程，从而得到y与x之间的幂律关系式。图2显示的是图1右图在双对数坐标下的图形，由于某些 因素的影响，图2前半部分的线性特性并不是很强，而在后半部分（对应于图1右图的尾部），则近乎为一直线，其斜率的负数就是幂指数。

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图2 双对数坐标下一个幂律分布的示意图，直线表示对图1右图尾部的线性拟合

　 　实际上，幂律分布[8]广泛存在于物理学、地球与行星科学、计算机科学、生物学、生态学、人口统计学与社会科学、经济与金融学等众多领域中，且表现形式 多种多样。在自然界与日常生活中，包括地震规模大小的分布[9](古登堡-里希特定律)、月球表面上月坑直径的分布[10]、行星间碎片大小的分布 [11]、太阳耀斑强度的分布[12]、计算机文件大小的分布[13]、战争规模的分布[14]、人类语言中单词频率的分布[5]、大多数国家姓氏的分布 [15]、科学家撰写的论文数的分布[16]、论文被引用的次数的分布[17]、网页被点击次数的分布[18]、书籍及唱片的销售册数或张数的分布 [19, 20]、每类生物中物种数的分布[21]、甚至电影所获得的奥斯卡奖项数的分布[22]等，都是典型的幂律分布。

　　以网页被点击次数的分布为例[23]，尽管中国向七千九百万网民提供的网站接近六十万个，但只有为数不多的网站，才拥有网民一次访问难以穷尽的丰富内容，拥有接纳许多人同时访问的足够带宽，进而有条件演化成热门网站，拥有极高的点击率，像新浪、搜狐、网易等门户网站。

　 　网页被点击次数的幂律分布其幂指数在0.60-1.03之间，而网站访问量的幂律分布其幂指数则接近1[24]。对于Pareto定律，在成熟市场中， 金融资产收益率的幂律分布其幂指数约等于3[25]。特别需要指出的是，一些幂律分布的幂指数带有一定的普适性，如月球表面的月坑，直径大于r的月坑总数 N(r)与r满足幂律关系，其幂指数D≈2.0，这一指数不仅对月球的月坑有效，甚至对火星和金星的火山口也有效[11]；还有一个是行星间碎片大小的分 布，其幂指数在2.0-2.1之间，这一区间不仅对陨石和小行星（如木星和火星轨道之间的小行星）这样的大碎片有效，而且对高速子弹打入岩石时所形成的小 碎片大小的分布也有效[11]；英文单词出现频率所满足的Zipf定律，不仅对报纸、《圣经》有效，而且对狄更斯的小说，莎士比亚的戏剧等也有效，甚至对 其它一些国家的语言也是有效的，且幂指数α均约等于1[26, 27]；情报学和科学学中有一个著名的公式，即洛特卡（Lotka）定律，它表明一定时期某一学科或主题内，撰写了x篇论文的作者数y(x)与x满足幂律 关系，不管学科或主题如何变化，其幂指数均在1.2-3.7之间，且大致按基础自然科学、技术科学、社会科学与人文科学的顺序递增[28]。

　 　幂律表现了一种很强的不平等性，对个人收入的分布来说这确实是一件很恐怖的事，但同时也说明了，单纯依据人均收入来衡量两个城市或国家的发展水平，并没 有多大的实际意义，必须还要提供一个衡量收入分布不均程度的参数――基尼系数[29, 30]，才能增强比较的可靠性。

　　统计物理学家习惯于把服从幂律分布的现象称为无标度现象，即，系统中个体的尺度相差悬殊，缺乏一个优选的规模。可以说，凡有生命的地方，有进化、有竞争的地方都会出现不同程度的无标度现象。

3 幂律分布研究：当前

　 　许多领域（像生物学、计算机科学）的进展都面临着要处理一些复杂系统问题[31]，自然界和社会中的系统的复杂性可归因于一个个交织的网络（像生态网、 因特网）的复杂性，通过这些复杂网络，系统的各个组成部分相互之间发生着各种线性的、非线性的作用。复杂网络[32-35]的研究应运而生，它是近年来刚 刚兴起的一个研究方向，隶属复杂性科学，教导我们从网络的观点来看待整个世界，甚至我们人类都可看成是复杂网络中的一个个小小的节点。钱学森[36]给出 了复杂网络的一个较严格的定义：具有自组织、自相似、吸引子、小世界、无标度中部分或全部性质的网络称为复杂网络。目前，这个新领域已聚集了一大批杰出的 物理学家、生物学家、计算机网络专家、数学家和社会学家。

　　从统计物理学来看，网络是一个包含了大量个体及个体之间相互作用的系统。近 年来在对复杂网络的研究过程中，科学家们亦发现了众多的幂律分布，虽然这些网络在结构及功能上是如此的千变万化，相差迥异。复杂网络中节点的度值k*相对 于它的概率P(k)满足幂律关系，且幂指数多在大于2小于3的范围内[31, 32]；这一现象是如此的普遍，如此地令人惊叹不已，以至于人们给具有这种性质的网络起了一个特别的名字――无标度网络[37]§。这里的无标度是指网络 缺乏一个特征度值（或平均度值），即节点度值的波动范围相当大。

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* 节点的度定义为与该节点相连接的节点的个数。
§ 可能地，Price[17] (Science, 1965)所研究的索引网络是第一个被发现的无标度网络。
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　 　无标度网络在自然界和现实生活中的实例举不胜举**，像Internet[38]、WWW[39, 40]这样的技术性网络，电子邮件网络[41]、电影演员合作网络[42]、引文关系网络[43]这样的社会性网络，甚至细胞代谢网络[44]、蛋白质调 控网络[45]、食物链网络[46]等之类的生物网，都是典型的无标度网络。在过去的40多年里，科学家们一直想当然地认为现实中的网络都是随机的，随机 图论[47]就是专门为了研究随机网络而发展起来的一门数学学科，但无标度特性的发现打破了这种构想。随机网络的度分布是泊松分布，度值比平均值高许多或 低许多的节点，都十分罕见，是一种高度"民主"的网络，而无标度网络的度分布则是幂律分布，节点度值相差悬殊，往往可以跨越几个数量级，是一种极端"专 制"的网络，二者之间有本质的区别。这两种网络的一个形象化的比较如图3[48]所示。

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** 存在一些指数型度分布的复杂网络[37]，如高速公路网，电力网。
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图3 具有相同节点数和边数的随机网络（左）和无标度网络（右）

　 　度分布满足幂律的无标度网络还有一个奇特的性质――"小世界"特性[49]，虽然WWW中的页面数已超过80亿，但平均来说，在WWW上只需点击19次 超链接，就可从一个网页到达任一其它页面。"小世界"现象在社会学上也称为"六度分离"，它来源于1967年，美国哈佛大学的社会心理学家Milgram 的一个实验[50-52]，这个实验证实，世界上任何两个人，不论他（她）是中国的藏民，非洲的难民，还是美国的政界高层，好莱坞的明星，甚至北极的爱斯 基摩人，美洲的土著印第安人，都可通过熟人找熟人的方式建立联系，而两者之间的平均最少"中介"数是6，如此看来，整个地球确实是一个小小的世界。

　 　图4[53]是Internet的拓扑图，它具有很强的自相似性，跟河流网之类的分形图非常类似。分形理论的创始人Mandelbrot[54]曾说 过，"当你看到一个非整数指数关系，就应想到分形。不过你应当小心从事"。可以说，幂律分布与分形、非线性、复杂性密切相关，它支配了所有自然演化的具有 自相似特性的无标度网络。无标度网络的度分布是一个非整数指数关系，这种网络的拓扑图呈现分形特征也在情理之中。近年来，物理工作者们日渐对无标度网络的 拓扑结构产生了浓厚的兴趣，并构建了多种物理定义，从不同的角度研究了无标度网络的分形维问题[55-57]。

　　简单性一向是现代自然 科学、特别是物理学的一条重要的指导原则[58]。许多科学家相信自然界的基本规律是简单的，爱因斯坦就是这种观点的突出代表者，他曾说过，"要使我们的 理论尽可能得简单――但不是更简单。"从普适简单的幂律，我们似乎可以说，大自然是如此的复杂，而支配它的物理定律却又是如此的简洁优雅。

4 幂律分布的形成机制

　 　如此广泛的幂律是怎样形成的呢？这是目前许多学者关注的焦点，毕竟一味地到处寻找幂律关系并没有多大的意义，而支配它形成的最根本的动力学原因才是最重 要的。从现象到本质的探索一直是物理学的使命，十几年来，或者几十年来，为了解释幂律分布的形成原因，科学家们提出了几种机制，包括增长与优先连接 [42, 59]、自组织临界[60, 61]、HOT理论[62, 63]、渗流模型[8,64-66]及一些随机过程[7, 8, 67]等。

　 　一些解释幂律形成原因的机制是相当复杂的，甚至动用了"临界现象理论"和"重正化群"[68, 69]等工具。其实，一些简单的代数方法――像"指数组合"[7, 8]、"变量替换"[70]――亦能产生幂律分布，比如，Miller[71]曾用"指数组合"的方法解释了英文单词频率的幂律分布，Reed和 Hughes[7]利用该机制，并结合随机过程，解释了城市人口分布、生物物种数分布等幂律分布。

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图4 Internet在自治系统层次上的拓扑图

　　4.1 优先连接

　 　Barabási与Albert针对复杂网络中普遍存在的幂律分布现象，提出了网络动态演化的BA模型[42, 59]，他们解释，成长性和优先连接性是无标度网络度分布呈现幂律的两个最根本的原因。所谓成长性是指网络节点数的增加，像Internet中自治系统或 路由器的添加，以及WWW中网站或网页的增加等，优先连接性是指新加入的节点总是优先选择与度值较高的节点相连，比如，新网站总是优先选择人们经常访问的 网站作为超链接。随着时间的演进，网络会逐渐呈现出一种"富者愈富，贫者愈贫"的现象。社会学家所说的"马太效应"[72]，《新约》圣经所说的"凡有 的，还要加给他，叫他有余"，同优先连接也有某种相通之处。

　　"优先连接性"的思想并不是BA模型的原创，早在1925年，Yule [73]在解释每类植物物种数的分布满足幂律分布的原因时就已经提出了类似的思想，虽然当时研究的对象不是复杂网络。1955年，Simon[74]对优 先连接性作了进一步深入的研究***，他对网络中可能存在的幂律不怎么感兴趣，但他列举了五种可以用他的理论解释的幂律分布：文献中单词频率的分布，科学 家撰写的科技文献数量的分布，城市人口的分布，收入的分布及每类生物中物种数的分布。

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*** 在Simon的工作之前，Champernowne[75]就提出了一个类似于"乘法过程"的数学模型，解释了个人收入分布的幂律现象。实际上，Simon的工作只是Champernowne模型的推广。
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　 　"优先连接"并不适用于所有出现幂律分布的情况，即便是对于某些无标度网络，用它解释幂律的成因也显得很不合理。以生态系统中的食物链为例，认为被捕食 者最有可能被猎物广泛的杂食性捕食者吃掉，确实是一件很荒唐的事。还有像Internet、航空网等网络，流量或容量的限制可以在一定程度上抑制优先连接 性，电影演员的合作网络中，节点（演员）的衰老或隐退也能起到类似的作用。

　　4.2 自组织临界

　　自 组织临界理论[61]是一个影响深远的理论，在复杂系统的研究领域中,该模型曾一直被认为是产生幂律分布的动力学原因，幂律亦可作为自组织临界的证据。它 认为，由大量相互作用的成分组成的系统会自然地向自组织临界态发展；当系统达到这种状态时，即使是很小的干扰事件也可能引起系统发生一系列灾变。布鲁克海 文实验室的Bak、加州大学圣巴巴拉分校的汤超和佐治亚理工学院的Wiesenfeld等人用著名的"沙堆模型"[61, 76]形象地说明了自组织临界态的形成和特点（如图5[76]）：设想在一平台上缓缓地添加沙粒，一个沙堆逐渐形成。开始时，由于沙堆平矮，新添加的沙粒 落下后不会滑得很远。但是，随着沙堆高度的增加，其坡度也不断增加，沙崩的规模也相应增大，但这些沙崩仍然是局部性的。到一定时候，沙堆的坡度会达到一个 临界值，这时，新添加一粒沙子（代表来自外界的微小干扰）就可能引起小到一粒或数粒沙子，大到涉及整个沙堆表面所有沙粒的沙崩。这时的沙堆系统处于"自组 织临界态",有趣的是，临界态时沙崩的大小与其出现的频率呈幂律关系。这里所谓的"自组织"是指该状态的形成主要是由系统内部各组成部分间的相互作用产 生，而不是由任何外界因素控制或主导所致，这是一个减熵有序化的过程；"临界态"是指系统处于一种特殊的敏感状态，微小的局部变化可以不断被放大、进而扩 延至整个系统。

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图5 "沙堆模型"

　 　幂律分布是自组织临界系统在混沌边缘，即从稳态过渡到混沌态的一个标志，利用它可以预测这类系统的相位及相变。自组织临界理论可以解释诸如火山爆发、山 体滑坡、岩层形成、日辉耀斑、物种灭绝、交通阻塞、以及金融市场中的幂律分布现象。这种理论的启示是小事件和大事件可能有相同的起因,这为地震、恐龙灭 绝、森林火灾等复杂大系统的突变提供了新的解释。以恐龙灭绝为例,古生物学家经过对化石的研究指出,这一重大事件不是经历了数万年或者几年,而是在20多 天的突变中"一朝覆灭"的。恐龙的灭绝可以被看作是处于临界状态下的生态系统发生的一次"大雪崩"。

　　4.3 HOT理论

　 　另一种解释幂律分布形成原因的重要理论是HOT[62, 63, 77]，由加州大学圣巴巴拉分校的Jean Carlson以及加州理工学院的John Doyle提出。他们宣称，对于由许多子系统连结成的复杂系统, 不管是自然演化还是人为设计的, 当该系统可以有效地容忍某些不确定因素时(具强健性)，将对其它未被考虑到的不确定因素变得更敏感。也就是说，强健性和敏感度具有相互递换的效果。这里的 不确定因素包含系统内部的不确定因素以及外在环境的干扰。以生态系统为例，如果它可以容忍气温变化、湿度、养分等巨幅变化，那么这生态系统却可能无法容忍 一些意料之外的小干扰，如基因突变、外来族群迁入、或新的病毒，这些干扰可能会造成生态环境的巨大改变。

　　当一复杂系统处于HOT状态 时，该系统将满足幂律，也就是说，全局性的优化过程可导致幂律分布：具有特征尺度的输入经过一个全局性的系统"产量"优化过程后，可产生具有幂律分布特性 的输出。全局性优化在生态系统、航空航天与汽车系统、林业系统、因特网、交通运输及电力系统中具有广泛的应用，HOT理论可以解释上述系统中出现的幂律分 布现象，比如可以解释林业系统中火灾规模所呈现的幂律分布。

　　4.4 随机过程

　　一些随机过程也可以产生幂律分布："随机行走"模型可以解释物种寿命所呈现的幂律分布[78]；"Yule过程"[21, 73]是一个生成幂律的比较通用的机制，通过调节它的某些参数，可以产生幂指数范围相当宽广的幂律分布，并可与实际观测值相一致。

　 　产生幂律分布的机制是相当多的，是否存在某个单一的、通用的理论可以解释所有的性质迥异的幂律分布呢？有一部分学者，特别是自组织临界理论的支持者，声 称他们的理论可以，但大多数科学家认为[79]，幂律分布是许多不同的过程或作用导致的结果，这就像经典力学，牛顿的经典力学固然很伟大，但它仅适用于宏 观低速的情形。

5 幂律分布的动力学影响

　　幂律分布的动力学影响主要是对复杂网络而言的。网络动力学性质的基本研究对象是动力学模型在不同网络上的性质与相应网络的度分布的联系，在一定程度上说，这是一种关于网络的结构与功能关系的研究。

　 　幂律特性的度分布对无标度网络的动力学性质有着极其深刻的影响。以疾病或病毒在网络中的传播这一物理过程为例，以前的基于规则网络及随机网络的研究表明 [80-82],疾病的传染强度存在一个阈值，只有传染强度大于这个阈值时,疾病才能在网络中长期存在,否则感染人数会呈指数衰减。但对无标度网络上传染 病模型的研究结果表明，不存在类似的阈值[83-86]，只要传染病发生，就将长时间存在下去，这一特性表明，要想在Internet这样的无标度网络上 彻底消灭病毒，即使是已知的病毒，也是不可能的[37]。

　　另外，度分布的幂律特性对网络的容错性和抗攻击能力也有很大的影响，对网络 的攻击分为随机攻击和选择性攻击两种类型[87]，分别称为网络的容错能力与抗攻击能力。研究表明[87, 88]，无标度网络具有很强的容错性，但是对基于节点度值的选择性攻击抗攻击能力相当差。比如对万维网或因特网中集散节点的攻击，有可能造成整个网络的瘫 痪，对于某些微生物来说，它们体内度值很高的蛋白质通常掌握着细胞的生死（如图6[37]所示）。度分布满足泊松分布的随机网络，其容错性和抗攻击能力则 是基本相当的[87]。可见，网络的结构稳定性是与网络的度分布特性紧密联系在一起的。

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图6 酵母菌体内蛋白质的相互作用关系图

　 　对于幂律分布对网络的其它动力学方面的影响，比如对网络上Ising模型[89, 90]、XY模型[91]、临界现象[92]及沙堆模型[93]等的影响，限于篇幅，本文不再赘述，有兴趣的读者可以参考相关文献。幂律分布对现实中无标 度网络的动力学性质影响深刻，这在相当程度上改变了我们对原有物理世界的看法，并进一步显示了幂律分布的重要性。

6 结束语

　 　幂律分布已有超过一百年的研究历史了，即使在现在，仍然是众多学科研究的热点。它那简洁优雅的形式，可以将许多似乎毫不相干的事物联系在一起，这种独特 的魅力吸引了一大批杰出的物理学家、生物学家、天文学家、地质学家、数学家和社会学家，并不断有新的研究者加入到该领域。但即便如此，要真正从本质上把握 驱动系统呈现幂律分布的物理过程与机制，仍然有许多试验或理论性的工作要做。另外，不同类型的幂律分布幂指数有很大的不同，究竟是什么原因导致了这种不 同？这仍然是一个尚未完全解决的问题。不过，我们相信，不久的将来，在众多科学家的共同努力下，人类最终将根本性地破解幂律分布之谜，为物理世界的简洁之 美再谱华章。

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Power Law, 幂律定律

自然界与社会生活中存在各种各样性质迥异的幂律分布现象。

1932 年,哈佛大学的语言学专家Zipf在研究英文单词出现的频率时,发现如果把单词出现的频率按由大到小的顺序排列,则每个单词出现的频率与它的名次的常数次幂存在简单的反比关系，这种分布就称为Zipf定律,它表明在英语单词中,只有极少数的词被经常使用,而绝大多数词很少被使用.实际上,包括汉语在内的许多国家的语言都有这种特点。

19世纪的意大利经济学家Pareto研究了个人收入的统计分布,发现少数人的收入要远多于大多数人的收入,提出了著名的80/20法则,即20%的人口占据了80%的 社会财富.个人收入X不小于某个特定值x的概率与x的常数次幂亦存在简单的反比关系，即为 Pareto定律。

Zipf 定律与 Pareto 定律都是简单的幂函数,我们称之为幂律分布;还有其它形式的幂律分布,像名次―规模分布,规模―概率分布,这四种形式在数学上是等价的。

幂律分布表现为一条斜率为幂指数的负数的直线,这一线性关系是判断给定的实例中随机变量是否满足幂律的依据。

实际上,幂律分布广泛存在于物理学,地球与行星科学,计算机科学,生物学,生态学,人口统计学与社会科学,经济与金融学等众多领域中,且表现形式多种多样.

在自然界与日常生活中,包括地震规模大小的分布(古登堡-里希特定律),月球表面上月坑直径的分布,行星间碎片大小的分布,太阳耀斑强度的分布,计算机文件大小的分布,战争规模的分布,人类语言中单词频率的分布,大多数国家姓氏的分布,科学家撰写的论文数的分布,论文被引用的次数的分布,网页被点击次数的分布,书籍及唱片的销售册数或张数的分布,每类生物中物种数的分布,甚至电影所获得的奥斯卡奖项数的分布等,都是典型的幂律分布。以网页被点击次数的分布为例,尽管中国向七千九百万网民提供的网站接近六十万个,但只有为数不多的网站,才拥有网民一次访问难以穷尽的丰富内容,拥有接纳许多人同时访问的足够带宽,进而有条件演化成热门网站,拥有极高的点击率,像新浪,搜狐,网易等门户网站。 网页被点击次数的幂律分布其幂指数在0.60- 1.03之间,而网站访问量的幂律分布其幂指数则接近1。

统计物理学家习惯于把服从幂律分布的现象称为无标度现象,即,系统中个体的尺度相差悬殊,缺乏一个优选的规模。可以说,凡有生命的地方,有进化,有竞争的地方都会出现
不同程度的无标度现象。

from [ http://mail.sxu.cn/blog/index.php?file=viewlog&uid=1451&id=209]

利用VS调试JAVASCRIPT

zz 利用VS调试JAVASCRIPT

(1) 打开IE --> Internet Options -- > Advanced ; 去掉"Disable Script Debugging" 上的选项

(2) 打开需要调试的页面

(3) 启动VS.Net IDE, 选择 TOOLS-Debug Process (Ctrl + Alt + P). 选择需要调试的IE进程。

(4) 点击Attach按钮，选择Script选项; 再OK关掉对话框

(5) 选择IDE菜单 DEBUGàWindowsàRunning Documents (Ctrl + Alt + N)

(6) 选择需要调试的Javascript并设置断点

(7) 回到IE页面，运行与JavaScript有关的功能，就会中断到断点上。这时可以观察相关的参数，就和其他VS.Net程序一样。

JS很有意思,可以动态增加成员变量,而且还可以以HASH表的形式访问..

SUBSTRING函数还是可以颠倒START和END 的..

Steve Jobs' Speech at Stanford

Transcript of Commencement Speech at Stanford given by Steve Jobs

Thank you. I'm honored to be with you today for your commencement from one of the finest universities in the world. Truth be told, I never graduated from college and this is the closest I've ever gotten to a college graduation.

Today I want to tell you three stories from my life. That's it. No big deal. Just three stories. The first story is about connecting the dots.

I dropped out of Reed College after the first six months but then stayed around as a drop-in for another eighteen months or so before I really quit. So why did I drop out? It started before I was born. My biological mother was a young, unwed graduate student, and she decided to put me up for adoption. She felt very strongly that I should be adopted by college graduates, so everything was all set for me to be adopted at birth by a lawyer and his wife, except that when I popped out, they decided at the last minute that they really wanted a girl. So my parents, who were on a waiting list, got a call in the middle of the night asking, "We've got an unexpected baby boy. Do you want him?" They said, "Of course." My biological mother found out later that my mother had never graduated from college and that my father had never graduated from high school. She refused to sign the final adoption papers. She only relented a few months later when my parents promised that I would go to college.

This was the start in my life. And seventeen years later, I did go to college, but I naïvely chose a college that was almost as expensive as Stanford, and all of my working-class parents' savings were being spent on my college tuition. After six months, I couldn't see the value in it. I had no idea what I wanted to do with my life, and no idea of how college was going to help me figure it out, and here I was, spending all the money my parents had saved their entire life. So I decided to drop out and trust that it would all work out OK. It was pretty scary at the time, but looking back, it was one of the best decisions I ever made. The minute I dropped out, I could stop taking the required classes that didn't interest me and begin dropping in on the ones that looked far more interesting.

It wasn't all romantic. I didn't have a dorm room, so I slept on the floor in friends' rooms. I returned Coke bottles for the five-cent deposits to buy food with, and I would walk the seven miles across town every Sunday night to get one good meal a week at the Hare Krishna temple. I loved it. And much of what I stumbled into by following my curiosity and intuition turned out to be priceless later on. Let me give you one example.

Reed College at that time offered perhaps the best calligraphy instruction in the country. Throughout the campus every poster, every label on every drawer was beautifully hand-calligraphed. Because I had dropped out and didn't have to take the normal classes, I decided to take a calligraphy class to learn how to do this. I learned about serif and sans-serif typefaces, about varying the amount of space between different letter combinations, about what makes great typography great. It was beautiful, historical, artistically subtle in a way that science can't capture, and I found it fascinating.

None of this had even a hope of any practical application in my life. But ten years later when we were designing the first Macintosh computer, it all came back to me, and we designed it all into the Mac. It was the first computer with beautiful typography. If I had never dropped in on that single course in college, the Mac would have never had multiple typefaces or proportionally spaced fonts, and since Windows just copied the Mac, it's likely that no personal computer would have them.

If I had never dropped out, I would have never dropped in on that calligraphy class and personals computers might not have the wonderful typography that they do.

Of course it was impossible to connect the dots looking forward when I was in college, but it was very, very clear looking backwards 10 years later. Again, you can't connect the dots looking forward. You can only connect them looking backwards, so you have to trust that the dots will somehow connect in your future. You have to trust in something--your gut, destiny, life, karma, whatever--because believing that the dots will connect down the road will give you the confidence to follow your heart, even when it leads you off the well-worn path, and that will make all the difference.

My second story is about love and loss. I was lucky. I found what I loved to do early in life. Woz and I started Apple in my parents' garage when I was twenty. We worked hard and in ten years, Apple had grown from just the two of us in a garage into a $2 billion company with over 4,000 employees. We'd just released our finest creation, the Macintosh, a year earlier, and I'd just turned thirty, and then I got fired. How can you get fired from a company you started? Well, as Apple grew, we hired someone who I thought was very talented to run the company with me, and for the first year or so, things went well. But then our visions of the future began to diverge, and eventually we had a falling out. When we did, our board of directors sided with him, and so at thirty, I was out, and very publicly out. What had been the focus of my entire adult life was gone, and it was devastating. I really didn't know what to do for a few months. I felt that I had let the previous generation of entrepreneurs down, that I had dropped the baton as it was being passed to me. I met with David Packard and Bob Noyce and tried to apologize for screwing up so badly. I was a very public failure and I even thought about running away from the Valley. But something slowly began to dawn on me. I still loved what I did. The turn of events at Apple had not changed that one bit. I'd been rejected but I was still in love. And so I decided to start over.

I didn't see it then, but it turned out that getting fired from Apple was the best thing that could have ever happened to me. The heaviness of being successful was replaced by the lightness of being a beginner again, less sure about everything. It freed me to enter one of the most creative periods in my life. During the next five years I started a company named NeXT, another company named Pixar and fell in love with an amazing woman who would become my wife. Pixar went on to create the world's first computer-animated feature film, "Toy Story," and is now the most successful animation studio in the world.

In a remarkable turn of events, Apple bought NeXT and I returned to Apple and the technology we developed at NeXT is at the heart of Apple's current renaissance, and Lorene and I have a wonderful family together.

I'm pretty sure none of this would have happened if I hadn't been fired from Apple. It was awful-tasting medicine but I guess the patient needed it. Sometimes life's going to hit you in the head with a brick. Don't lose faith. I'm convinced that the only thing that kept me going was that I loved what I did. You've got to find what you love, and that is as true for work as it is for your lovers. Your work is going to fill a large part of your life, and the only way to be truly satisfied is to do what you believe is great work, and the only way to do great work is to love what you do. If you haven't found it yet, keep looking, and don't settle. As with all matters of the heart, you'll know when you find it, and like any great relationship it just gets better and better as the years roll on. So keep looking. Don't settle.

My third story is about death. When I was 17 I read a quote that went something like "If you live each day as if it was your last, someday you'll most certainly be right." It made an impression on me, and since then, for the past 33 years, I have looked in the mirror every morning and asked myself, "If today were the last day of my life, would I want to do what I am about to do today?" And whenever the answer has been "no" for too many days in a row, I know I need to change something. Remembering that I'll be dead soon is the most important thing I've ever encountered to help me make the big choices in life, because almost everything--all external expectations, all pride, all fear of embarrassment or failure--these things just fall away in the face of death, leaving only what is truly important. Remembering that you are going to die is the best way I know to avoid the trap of thinking you have something to lose. You are already naked. There is no reason not to follow your heart.

About a year ago, I was diagnosed with cancer. I had a scan at 7:30 in the morning and it clearly showed a tumor on my pancreas. I didn't even know what a pancreas was. The doctors told me this was almost certainly a type of cancer that is incurable, and that I should expect to live no longer than three to six months. My doctor advised me to go home and get my affairs in order, which is doctors' code for "prepare to die." It means to try and tell your kids everything you thought you'd have the next ten years to tell them, in just a few months. It means to make sure that everything is buttoned up so that it will be as easy as possible for your family. It means to say your goodbyes.

I lived with that diagnosis all day. Later that evening I had a biopsy where they stuck an endoscope down my throat, through my stomach into my intestines, put a needle into my pancreas and got a few cells from the tumor. I was sedated but my wife, who was there, told me that when they viewed the cells under a microscope, the doctor started crying, because it turned out to be a very rare form of pancreatic cancer that is curable with surgery. I had the surgery and, thankfully, I am fine now.

This was the closest I've been to facing death, and I hope it's the closest I get for a few more decades. Having lived through it, I can now say this to you with a bit more certainty than when death was a useful but purely intellectual concept. No one wants to die, even people who want to go to Heaven don't want to die to get there, and yet, death is the destination we all share. No one has ever escaped it. And that is as it should be, because death is very likely the single best invention of life. It's life's change agent; it clears out the old to make way for the new. right now, the new is you. But someday, not too long from now, you will gradually become the old and be cleared away. Sorry to be so dramatic, but it's quite true. Your time is limited, so don't waste it living someone else's life. Don't be trapped by dogma, which is living with the results of other people's thinking. Don't let the noise of others' opinions drown out your own inner voice, heart and intuition. They somehow already know what you truly want to become. Everything else is secondary.

When I was young, there was an amazing publication called The Whole Earth Catalogue, which was one of the bibles of my generation. It was created by a fellow named Stuart Brand not far from here in Menlo Park, and he brought it to life with his poetic touch. This was in the late Sixties, before personal computers and desktop publishing, so it was all made with typewriters, scissors, and Polaroid cameras. it was sort of like Google in paperback form thirty-five years before Google came along. I was idealistic, overflowing with neat tools and great notions. Stuart and his team put out several issues of the The Whole Earth Catalogue, and then when it had run its course, they put out a final issue. It was the mid-Seventies and I was your age. On the back cover of their final issue was a photograph of an early morning country road, the kind you might find yourself hitchhiking on if you were so adventurous. Beneath were the words, "Stay hungry, stay foolish." It was their farewell message as they signed off. "Stay hungry, stay foolish." And I have always wished that for myself, and now, as you graduate to begin anew, I wish that for you. Stay hungry, stay foolish.

Thank you all, very much.


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"Stay hungry, stay foolish"------Steve Jobs

" Do one thing,do it very well,and enjoy it."